Expected value of a continuous function. The random variable (X) has pdf (f_X(x) = 2x) for (0 le x le 1) and zero otherwise. (a) Compute (E[X]), (E[X^2]) and (Var(X)). Show that (Var(X) = E[X^2] - (E[X])^2). (b) Define the function (g(x) = frac{1}{1+x}). Find (E[g(X)]) by integrating (g(x) f_X(x)) over the support of (X). (c) If (Y = sqrt{X}), compute (E[Y]). Discuss how the nonlinear transformation of (X) affects the mean compared with (E[X])